### Abstract

We denote by h_{D} the class number and by pD the Ono number of the imaginary quadratic fields Q(√-D). Sairaiji-Shimizu [2] showed that there are infinitely many imaginary quadratic fields such that the inequality h_{D} > c^{pD} holds for any real number. On the other hand we have the possibility that h_{D} 5 c^{pD} holds for infinitely many imaginary quadratic fields for the same real number c. In this paper, given a real number c, we consider whether h_{D} 5 c^{pD} holds for infinitely many imaginary quadratic fields or not.

Original language | English |
---|---|

Pages (from-to) | 46-48 |

Number of pages | 3 |

Journal | Proceedings of the Japan Academy Series A: Mathematical Sciences |

Volume | 85 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2009 |

Externally published | Yes |

### Keywords

- Class number
- Ono number

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Inequalities of Ono numbers and class numbers associated to imaginary quadratic fields.** / Shimizu, Kenichi.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Inequalities of Ono numbers and class numbers associated to imaginary quadratic fields

AU - Shimizu, Kenichi

PY - 2009

Y1 - 2009

N2 - We denote by hD the class number and by pD the Ono number of the imaginary quadratic fields Q(√-D). Sairaiji-Shimizu [2] showed that there are infinitely many imaginary quadratic fields such that the inequality hD > cpD holds for any real number. On the other hand we have the possibility that hD 5 cpD holds for infinitely many imaginary quadratic fields for the same real number c. In this paper, given a real number c, we consider whether hD 5 cpD holds for infinitely many imaginary quadratic fields or not.

AB - We denote by hD the class number and by pD the Ono number of the imaginary quadratic fields Q(√-D). Sairaiji-Shimizu [2] showed that there are infinitely many imaginary quadratic fields such that the inequality hD > cpD holds for any real number. On the other hand we have the possibility that hD 5 cpD holds for infinitely many imaginary quadratic fields for the same real number c. In this paper, given a real number c, we consider whether hD 5 cpD holds for infinitely many imaginary quadratic fields or not.

KW - Class number

KW - Ono number

UR - http://www.scopus.com/inward/record.url?scp=84904399525&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84904399525&partnerID=8YFLogxK

U2 - 10.3792/pjaa.85.46

DO - 10.3792/pjaa.85.46

M3 - Article

AN - SCOPUS:84904399525

VL - 85

SP - 46

EP - 48

JO - Proceedings of the Japan Academy Series A: Mathematical Sciences

JF - Proceedings of the Japan Academy Series A: Mathematical Sciences

SN - 0386-2194

IS - 4

ER -